Sequence Alignment

Complete DNA Sequences

More than 1000 complete genomes have been sequenced

Comparative Genomics

Evolution at the DNA level

…ACGGTGCAGTTACCA…

…AC----CAGTCCACCA…

Mutation

SEQUENCE EDITS

REARRANGEMENTS

Deletion

InversionTranslocationDuplication

Evolutionary Rates

OK

OK

OK

X

X

Still OK?

next generation

Sequence conservation implies function

Alignment is the key to• Finding important regions• Determining function• Uncovering evolutionary events

Sequence Alignment

Sequence Alignment

-AGGCTATCACCTGACCTCCAGGCCGA--TGCCC---TAG-CTATCAC--GACCGC--GGTCGATTTGCCCGAC

DefinitionGiven two strings x = x1x2...xM, y = y1y2…yN,

an alignment is an assignment of gaps to positions0,…, N in x, and 0,…, N in y, so as to line up each

letter in one sequence with either a letter, or a gapin the other sequence

AGGCTATCACCTGACCTCCAGGCCGATGCCCTAGCTATCACGACCGCGGTCGATTTGCCCGAC

What is a good alignment?

AGGCTAGTT, AGCGAAGTTT

AGGCTAGTT- 6 matches, 3 mismatches, 1 gap

AGCGAAGTTT

AGGCTA-GTT- 7 matches, 1 mismatch, 3 gaps

AG-CGAAGTTT

AGGC-TA-GTT- 7 matches, 0 mismatches, 5 gaps

AG-CG-AAGTTT

Scoring Function

• Sequence edits:AGGCCTC

Mutations AGGACTC

Insertions AGGGCCTC

Deletions AGG . CTC

Scoring Function:Match: +mMismatch: -sGap: -d

Score F = (# matches) m - (# mismatches) s – (#gaps) d

Alternative definition:

minimal edit distance

“Given two strings x, y,find minimum # of edits (insertions, deletions,

mutations) to transform one string to the other”

How do we compute the best alignment?

AGTGCCCTGGAACCCTGACGGTGGGTCACAAAACTTCTGGA

AGTGACCTGGGAAGACCCTGACCCTGGGTCACAAAACTC

Too many possible alignments:

>> 2N

(exercise)

Alignment is additive

Observation:The score of aligning x1……xM

y1……yN

is additive

Say that x1…xi xi+1…xM aligns to y1…yj yj+1…yN

The two scores add up:

F(x[1:M], y[1:N]) = F(x[1:i], y[1:j]) + F(x[i+1:M], y[j+1:N])

Dynamic Programming

• There are only a polynomial number of subproblems Align x1…xi to y1…yj

• Original problem is one of the subproblems Align x1…xM to y1…yN

• Each subproblem is easily solved from smaller subproblems We will show next

• Then, we can apply Dynamic Programming!!!

Let F(i, j) = optimal score of aligning

x1……xi

y1……yj

F is the DP “Matrix” or “Table”

“Memoization”

Dynamic Programming (cont’d)

Notice three possible cases:

1. xi aligns to yj

x1……xi-1 xi

y1……yj-1 yj

2. xi aligns to a gapx1……xi-1 xi

y1……yj -

3. yj aligns to a gapx1……xi -y1……yj-1 yj

m, if xi = yj

F(i, j) = F(i – 1, j – 1) + -s, if

not

F(i, j) = F(i – 1, j) – d

F(i, j) = F(i, j – 1) – d

Dynamic Programming (cont’d)

How do we know which case is correct?

Inductive assumption:F(i, j – 1), F(i – 1, j), F(i – 1, j – 1) are optimal

Then, F(i – 1, j – 1) + s(xi, yj)

F(i, j) = max F(i – 1, j) – d F(i, j – 1) – d

Where s(xi, yj) = m, if xi = yj; -s, if not

G -

A G T A

0 -1 -2 -3 -4

A -1 1 0 -1 -2

T -2 0 0 1 0

A -3 -1 -1 0 2

F(i,j) i = 0 1 2 3 4

Example

x = AGTA m = 1y = ATA s = -1

d = -1

j = 0

1

2

3

F(1, 1) = max{F(0,0) + s(A, A), F(0, 1) – d, F(1, 0) – d} =

max{0 + 1, -1 – 1, -1 – 1} = 1

AA

TT

AA

Procedure to output Alignment

• Follow the backpointers

• When diagonal,OUTPUT xi, yj

• When up,OUTPUT yj

• When left,OUTPUT xi

The Needleman-Wunsch Matrix

x1 ……………………………… xMy1 …

……

……

……

……

……

… y

N

Every nondecreasing path

from (0,0) to (M, N)

corresponds to an alignment of the two sequences

An optimal alignment is composed of optimal subalignments

The Needleman-Wunsch Algorithm

1. Initialization.a. F(0, 0) = 0b. F(0, j) = - j dc. F(i, 0) = - i d

2. Main Iteration. Filling-in partial alignmentsa. For each i = 1……M

For each j = 1……N F(i – 1,j – 1) + s(xi, yj)

[case 1]F(i, j) = max F(i – 1, j) – d [case

2] F(i, j – 1) – d [case

3]

DIAG, if [case 1]Ptr(i, j) = LEFT, if [case 2]

UP, if [case 3]

3. Termination. F(M, N) is the optimal score, andfrom Ptr(M, N) can trace back optimal alignment

Performance

• Time:O(NM)

• Space:O(NM)

• Later we will cover more efficient methods

A variant of the basic algorithm:

• Maybe it is OK to have an unlimited # of gaps in the beginning and end:

----------CTATCACCTGACCTCCAGGCCGATGCCCCTTCCGGC ||||||| |||| | || ||GCGAGTTCATCTATCAC--GACCGC--GGTCG--------------

• Then, we don’t want to penalize gaps in the ends

Different types of overlaps

Example:2 overlapping“reads” from a sequencing project – recall Lecture 1

Example:Search for a mouse genewithin a human chromosome

The Overlap Detection variant

Changes:

1. InitializationFor all i, j,

F(i, 0) = 0F(0, j) = 0

2. Termination maxi F(i, N)

FOPT = max maxj F(M, j)

x1 ……………………………… xM

y1 …

……

……

……

……

……

…

yN

The local alignment problem

Given two strings x = x1……xM,

y = y1……yN

Find substrings x’, y’ whose similarity (optimal global alignment value)is maximum

x = aaaacccccggggttay = ttcccgggaaccaacc

Why local alignment – examples

• Genes are shuffled between genomes

• Portions of proteins (domains) are often conserved

Cross-species genome similarity

• 98% of genes are conserved between any two mammals• >70% average similarity in protein sequence

hum_a : GTTGACAATAGAGGGTCTGGCAGAGGCTC--------------------- @ 57331/400001mus_a : GCTGACAATAGAGGGGCTGGCAGAGGCTC--------------------- @ 78560/400001rat_a : GCTGACAATAGAGGGGCTGGCAGAGACTC--------------------- @ 112658/369938fug_a : TTTGTTGATGGGGAGCGTGCATTAATTTCAGGCTATTGTTAACAGGCTCG @ 36008/68174 hum_a : CTGGCCGCGGTGCGGAGCGTCTGGAGCGGAGCACGCGCTGTCAGCTGGTG @ 57381/400001mus_a : CTGGCCCCGGTGCGGAGCGTCTGGAGCGGAGCACGCGCTGTCAGCTGGTG @ 78610/400001rat_a : CTGGCCCCGGTGCGGAGCGTCTGGAGCGGAGCACGCGCTGTCAGCTGGTG @ 112708/369938fug_a : TGGGCCGAGGTGTTGGATGGCCTGAGTGAAGCACGCGCTGTCAGCTGGCG @ 36058/68174

hum_a : AGCGCACTCTCCTTTCAGGCAGCTCCCCGGGGAGCTGTGCGGCCACATTT @ 57431/400001mus_a : AGCGCACTCG-CTTTCAGGCCGCTCCCCGGGGAGCTGAGCGGCCACATTT @ 78659/400001rat_a : AGCGCACTCG-CTTTCAGGCCGCTCCCCGGGGAGCTGCGCGGCCACATTT @ 112757/369938fug_a : AGCGCTCGCG------------------------AGTCCCTGCCGTGTCC @ 36084/68174 hum_a : AACACCATCATCACCCCTCCCCGGCCTCCTCAACCTCGGCCTCCTCCTCG @ 57481/400001mus_a : AACACCGTCGTCA-CCCTCCCCGGCCTCCTCAACCTCGGCCTCCTCCTCG @ 78708/400001rat_a : AACACCGTCGTCA-CCCTCCCCGGCCTCCTCAACCTCGGCCTCCTCCTCG @ 112806/369938fug_a : CCGAGGACCCTGA------------------------------------- @ 36097/68174

“atoh” enhancer in human, mouse, rat, fugu fish

The Smith-Waterman algorithm

Idea: Ignore badly aligning regions

Modifications to Needleman-Wunsch:

Initialization: F(0, j) = F(i, 0) = 0

0

Iteration: F(i, j) = max F(i – 1, j) – d

F(i, j – 1) – d

F(i – 1, j – 1) + s(xi, yj)

The Smith-Waterman algorithm

Termination:

1. If we want the best local alignment…

FOPT = maxi,j F(i, j)

Find FOPT and trace back

2. If we want all local alignments scoring > t

?? For all i, j find F(i, j) > t, and trace back?

Complicated by overlapping local alignments

Waterman–Eggert ’87: find all non-overlapping local alignments with minimal recalculation of the DP matrix

Scoring the gaps more accurately

Current model:

Gap of length nincurs penalty nd

However, gaps usually occur in bunches

Convex gap penalty function:

(n):for all n, (n + 1) - (n) (n) - (n – 1)

(n)

(n)

Convex gap dynamic programming

Initialization: same

Iteration:

F(i – 1, j – 1) + s(xi, yj)

F(i, j) = max maxk=0…i-1F(k, j) – (i – k)

maxk=0…j-1F(i, k) – (j – k)

Termination: same

Running Time: O(N2M) (assume N>M)

Space: O(NM)

Compromise: affine gaps

(n) = d + (n – 1)e | | gap gap open extend

To compute optimal alignment,

At position i, j, need to “remember” best score if gap is open best score if gap is not open

F(i, j): score of alignment x1…xi to y1…yj

if xi aligns to yj

G(i, j): score if xi aligns to a gap after yj

H(i, j): score if yj aligns to a gap after xi

V(i, j) = best score of alignment x1…xi to y1…yj

de

(n)

Needleman-Wunsch with affine gaps

Why do we need matrices F, G, H?

• xi aligns to yj

x1……xi-1 xi xi+1

y1……yj-1 yj -

• xi aligns to a gap after yj

x1……xi-1 xi xi+1

y1……yj …- -

Add -d

Add -e

G(i+1, j) = F(i, j) – d

G(i+1, j) = G(i, j) – e

Because, perhaps

G(i, j) < V(i, j)

(it is best to align xi to yj if we were aligningonly x1…xi to y1…yj and not the rest of x, y),

but on the contrary

G(i, j) – e > V(i, j) – d

(i.e., had we “fixed” our decision that xi alignsto yj, we could regret it at the next step whenaligning x1…xi+1 to y1…yj)

Needleman-Wunsch with affine gaps

Initialization: V(i, 0) = d + (i – 1)eV(0, j) = d + (j – 1)e

Iteration:V(i, j) = max{ F(i, j), G(i, j), H(i, j) }

F(i, j) = V(i – 1, j – 1) + s(xi, yj)

V(i – 1, j) – d G(i, j) = max

G(i – 1, j) – e

V(i, j – 1) – d H(i, j) = max

H(i, j – 1) – e

Termination: V(i, j) has the best alignmentTime?

Space?

To generalize a bit…

… think of how you would compute optimal alignment with this gap function

….in time O(MN)

(n)

Bounded Dynamic Programming

Assume we know that x and y are very similar

Assumption: # gaps(x, y) < k(N)

xi Then, | implies | i – j | < k(N)

yj

We can align x and y more efficiently:

Time, Space: O(N k(N)) << O(N2)

Bounded Dynamic Programming

Initialization:

F(i,0), F(0,j) undefined for i, j > k

Iteration:

For i = 1…M

For j = max(1, i – k)…min(N, i+k)

F(i – 1, j – 1)+ s(xi, yj)

F(i, j) = max F(i, j – 1) – d, if j > i – k(N)

F(i – 1, j) – d, if j < i + k(N)

Termination: same

Easy to extend to the affine gap case

x1 ………………………… xM

y1 …

……

……

……

……

…

yN

k(N)

Linear-Space Alignment

Subsequences and Substrings

Definition A string x’ is a substring of a string x,if x = ux’v for some prefix string u and suffix string v

(similarly, x’ = xi…xj, for some 1 i j |x|)

A string x’ is a subsequence of a string xif x’ can be obtained from x by deleting 0 or more letters

(x’ = xi1…xik, for some 1 i1 … ik |x|)

Note: a substring is always a subsequence

Example: x = abracadabray = cadabr; substringz = brcdbr; subseqence, not substring

Hirschberg’s algortihm

Given a set of strings x, y,…, a common subsequence is a string u that is a subsequence of all strings x, y, …

• Longest common subsequence Given strings x = x1 x2 … xM, y = y1 y2 … yN, Find longest common subsequence u = u1 … uk

• Algorithm:F(i – 1, j)

• F(i, j) = max F(i, j – 1)F(i – 1, j – 1) + [1, if xi = yj; 0 otherwise]

• Ptr(i, j) = (same as in N-W)

• Termination: trace back from Ptr(M, N), and prepend a letter to u whenever • Ptr(i, j) = DIAG and F(i – 1, j – 1) < F(i, j)

• Hirschberg’s original algorithm solves this in linear space

F(i,j)

Introduction: Compute optimal score

It is easy to compute F(M, N) in linear space

Allocate ( column[1] )

Allocate ( column[2] )

For i = 1….M

If i > 1, then:

Free( column[ i – 2 ] )

Allocate( column[ i ] )

For j = 1…N

F(i, j) = …

Linear-space alignment

To compute both the optimal score and the optimal alignment:

Divide & Conquer approach:

Notation:

xr, yr: reverse of x, y

E.g. x = accgg;

xr = ggcca

Fr(i, j): optimal score of aligning xr1…xr

i & yr1…yr

j

same as aligning xM-i+1…xM & yN-j+1…yN

Linear-space alignment

Lemma: (assume M is even)

F(M, N) = maxk=0…N( F(M/2, k) + Fr(M/2, N – k) )

x

y

M/2

k*

F(M/2, k) Fr(M/2, N – k)

Example:ACC-GGTGCCCAGGACTG--CATACCAGGTG----GGACTGGGCAG

k* = 8

Linear-space alignment

• Now, using 2 columns of space, we can compute

for k = 1…M, F(M/2, k), Fr(M/2, N – k)

PLUS the backpointers

x1 … xM/2

y1

xM

yN

x1 … xM/2+1 xM

…

y1

yN

…

Linear-space alignment

• Now, we can find k* maximizing F(M/2, k) + Fr(M/2, N-k)

• Also, we can trace the path exiting column M/2 from k*

k*

k*+1

0 1 …… M/2 M/2+1 …… M M+1

Linear-space alignment

• Iterate this procedure to the left and right!

N-k*

M/2M/2

k*

Linear-space alignment

Hirschberg’s Linear-space algorithm:

MEMALIGN(l, l’, r, r’): (aligns xl…xl’ with yr…yr’)

1. Let h = (l’-l)/22. Find (in Time O((l’ – l) (r’ – r)), Space O(r’ – r))

the optimal path, Lh, entering column h – 1, exiting column hLet k1 = pos’n at column h – 2 where Lh enters

k2 = pos’n at column h + 1 where Lh exits

3. MEMALIGN(l, h – 2, r, k1)

4. Output Lh

5. MEMALIGN(h + 1, l’, k2, r’)

Top level call: MEMALIGN(1, M, 1, N)

Linear-space alignment

Time, Space analysis of Hirschberg’s algorithm: To compute optimal path at middle column,

For box of size M N,Space: 2NTime: cMN, for some constant c

Then, left, right calls cost c( M/2 k* + M/2 (N – k*) ) = cMN/2

All recursive calls cost Total Time: cMN + cMN/2 + cMN/4 + ….. = 2cMN = O(MN)

Total Space: O(N) for computation, O(N + M) to store the optimal alignment

Heuristic Local Alignerers

1. The basic indexing & extension technique

2. Indexing: techniques to improve sensitivityPairs of Words, Patterns

3. Systems for local alignment

Indexing-based local alignment

Dictionary:

All words of length k (~10)

Alignment initiated between words of alignment score T

(typically T = k)

Alignment:

Ungapped extensions until score

below statistical threshold

Output:

All local alignments with score

> statistical threshold

……

……

query

DB

query

scan

Indexing-based local alignment—Extensions

A C G A A G T A A G G T C C A G T

C

T

G

A

T

C C

T

G

G

A

T

T

G C

G

A

Gapped extensions until threshold

• Extensions with gaps until score < C below best score so far

Output:

GTAAGGTCCAGTGTTAGGTC-AGT

Sensitivity-Speed Tradeoff

long words(k = 15)

short words(k = 7)

Sensitivity Speed

Kent WJ, Genome Research 2002

Sens.

Speed

X%

Sensitivity-Speed Tradeoff

Methods to improve sensitivity/speed

1. Using pairs of words

2. Using inexact words

3. Patterns—non consecutive positions

……ATAACGGACGACTGATTACACTGATTCTTAC……

……GGCACGGACCAGTGACTACTCTGATTCCCAG……

……ATAACGGACGACTGATTACACTGATTCTTAC……

……GGCGCCGACGAGTGATTACACAGATTGCCAG……

TTTGATTACACAGAT T G TT CAC G

Measured improvement

Kent WJ, Genome Research 2002

Non-consecutive words—Patterns

Patterns increase the likelihood of at least one match within a long conserved region

3 common

5 common

7 common

Consecutive Positions Non-Consecutive Positions

6 common

On a 100-long 70% conserved region: Consecutive Non-consecutive

Expected # hits: 1.07 0.97Prob[at least one hit]: 0.30 0.47

Advantage of Patterns

11 positions

11 positions

10 positions

Multiple patterns

• K patterns Takes K times longer to scan Patterns can complement one another

• Computational problem: Given: a model (prob distribution) for homology between two regions Find: best set of K patterns that maximizes Prob(at least one match)

TTTGATTACACAGAT T G TT CAC G T G T C CAG TTGATT A G

Buhler et al. RECOMB 2003Sun & Buhler RECOMB 2004

How long does it take to search the query?

Variants of BLAST

• NCBI BLAST: search the universe http://www.ncbi.nlm.nih.gov/BLAST/• MEGABLAST: http://genopole.toulouse.inra.fr/blast/megablast.html

Optimized to align very similar sequences• Works best when k = 4i 16• Linear gap penalty

• WU-BLAST: (Wash U BLAST) http://blast.wustl.edu/ Very good optimizations Good set of features & command line arguments

• BLAT http://genome.ucsc.edu/cgi-bin/hgBlat Faster, less sensitive than BLAST Good for aligning huge numbers of queries

• CHAOS http://www.cs.berkeley.edu/~brudno/chaos Uses inexact k-mers, sensitive

• PatternHunter http://www.bioinformaticssolutions.com/products/ph/index.php Uses patterns instead of k-mers

• BlastZ http://www.psc.edu/general/software/packages/blastz/ Uses patterns, good for finding genes

• Typhon http://typhon.stanford.edu Uses multiple alignments to improve sensitivity/speed tradeoff